1. Limits and Continuity

Limits and Infinity

Find the limits in Exercises 37–46.

x⁴ + x³

lim -----------------

x→∞ 12x³ + 128

Limits of quotients

Find the limits in Exercises 23–42.

limx→−1 (√(x² + 8) − 3) / (x + 1)

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=cos x+2√x / √x.

Determine the following limits.

lim x→3 x^4 − 81 / x − 3

How is lim x🠂0 sin x/x used in this section?

82. Use the definitions of the hyperbolic functions to find each of the following limits.

f. lim(x→∞) coth x

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (sin 3x) / x

Evaluate each limit. 

$\underset{x\to \pi }{\lim }\frac{{\cos }^2\left(x\right)+3\cos \left(x\right)+2}{{\cos }^{}\left(x\right)+1}$

110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.

a. f(x) = 3^(-x), g(x) = 2^(-x)

Even function limits Suppose f is an even function where lim x→1^− f(x)=5 and lim x→1^+ f(x)=6. Find lim x→−1^− f(x) and limx→−1^+ f(x).

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 5x) / x

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim t→2+ |2t − 4|t^2 − 4

Determine the following limits. 

lim x→−∞ (5 + 100/x + sin⁴ x³ / x²)

Limits and Infinity

Find the limits in Exercises 37–46.

x²/³ + x⁻¹

lim --------------------

x→∞ x²/³ + cos²x

9. True, or false? As x→∞,

g. ln(x) = o(ln(2x))